Exchange Scattering of an Electron by the Hydrogen Atom

Abstract

where we have made the substitution v=| (zZe 2 /E)u and are using the dimensionless constants ^=[2£/( 2 Ze 2 )>=tan(i$ 0 ), (15) p=a8(hc/e 2 )z-A Z-2 (E/hcy = 2.68X10 39 as-4 Z-2 E Me v 3 , (16) where a is in cm 3 .Hardmeier 7 has given a formula for 8 at small scattering angles and has evaluated it explicitly for large scattering angles for 0= 1, 2, 4, 6. We have evaluated 8If we evaluate this as a power series in /3, we obtain for backward scattering • -(0) = 1-0.2290-O.O3540 8 .For /3=0.2, this gives us a 4.7 percent deviation from pure Coulomb scattering in the backward direction. For the deuteron z=l and from reference 1, we have 8 8 The value of a is taken from Eqs. (13) and (21) of reference 1 by noting that for the scattering problem, we must average over T HE analysis of exchange scattering was originally made by Oppenheimer, 1 and an entirely different treatment of this problem which has become standard was given by Mott and Massey. 2 The reason for the superiority of the latter method is that the general matrix element for the exchange scattered amplitude is identified, while Oppenheimer s solution is an approximate one from the outset so that the possibility for improved estimates is automatically ruled out. HardmeierV results for small scattering angles.In the case of head on collisions, <T P /<T C reduces to [1+ (d8/d$o)l~2', also, as £->0, we have a=0.56X10-39 cm 3 . Then, applying Eqs. (16) and (18) to some examples where the nuclear effects should be comparatively small 2 -9 and hence our treatment be valid, we find for 8-Mev deuterons scattered by 83Bi where /3=0.112 a deviation of 2.7 percent in the backward direction, and for 10-Mev deuterons scattered by 92U where 0=0.177 a deviation of 4.2 percent in the backward direction. the magnetic quantum number so that (a)to = ass-\r2aDD~ass = 0.56X10-39 cm 3 . 9 D. C. Peaslee, Phys. Rev. 74, 1001 (1948); C. J. Mullin and E. Guth, Phys. Rev. 82, 141 (1951). assumption that the usual stationary state solution from which the direct scattered amplitude is derived has the asymptotic formwhere ri and r 2 refer to the primary and hydrogenic electrons, respectively. (This labeling will prevail throughout the present paper.) The <p y s are Coulomb functions; the sum includes integration over continuum states. Mott and Massey point out that no proof of this assumption has been given but that one should be possible. It is the purpose of this paper to provide suchThe theory of scattering of electrons by hydrogen has been re-examined with the objective of identifying the matrix element for the exchange scattered amplitude from the same integral equation which provides direct scattered amplitudes. The theory of Mott and Massey is verified, but it is demonstrated that their result contains contributions from the incident wave which must be removed before computing exchange amplitudes. The result is a theoretical justification for the Oppenheimer (prior) matrix element. 1168SAUL ALTSHULER a proof 3 and to discuss other consequences arising from the analysis. Unless...

Scattering Theory of Waves and Particles

“…For a specific discussion of the occurrence of the incoming wave state as a final state and the outgoing wave state as the initial state, see Breit and Bethe (1954) and also S. Altshuler (1956). For studies of the off-energy-shell T matrix, see J.…”

Scattering of an Electron by an Atom and Rearrangement Collisions

“…In the preceding treatment of Trainor and Wu, the total (direct and exchange) scattering is obtained together in (10) and (14). But a proof of (20) has been given by Moses (1953) and by Altshuler (1953). The latter method is particularly interesting since it call be extended to the case involving nonidentical particles (for example, the collision of a negative meson and a hydrogen atom), and siilce its equivalence to the result (10) of Traillor and Wu can be readily shown.…”

“…Finally, it is of interest to show that a theorem essentially equivalent t o that of Altshuler (1953) in $11 can be proved in this theory of transitions. The initial condition on 9 , such as (33), together with the time-dependent Schrodinger equation (29), completely determines the y5 a t subsequent times.…”

“…A rigorous but incomplete treatment of the problem was given by Mott and Rlassey (1933), and the conlpletion of this treatment has been the subject of a few recent studies (Trainor and \Vu 1953;Altshuler 1953;Moses 1953). Both because some rather important points are not sufficiently recognized and because of the possibility of extending the treatment of the problem to other rearrangement collisions of which the "exchange" scattering of an electron by a hydrogen atom is perhaps the simplest example, we shall begin with a brief resume of the treatments of Trainor and Wu (1953) and of Altshuler (1953) and a demonstration that these two treatments are exactly equivalent.…”

“…This treatment leads to a different result from that of the usual treatment even in what corresponds to the first "Born approximation". The purpose of the present work is (i) to start from the rigorous treatment of Altshuler (1953) of the "exchange" scattering of an electron by a hydrogen atom, which is shown to be equivalent to a much simpler treatment by Trainor and Wu (1953) based on the symmetry property of the system in the two electrons, (ii) to extend Altshuler's treatment to other rearrangement collisions, (iii) to justify the usual perturbation treatment involving nonorthogonal states, and (iv) to show that Dirac's recent formulati011 and result, while apparently difierent from the usual treatment, are in fact completely identical with the usual ones when properly carried out and interpreted.…”